Question: All of the 4th grade teachers and students from Covington went on a field trip to an art museum. Tickets were $$6.50$ each for teachers and $$2.00$ each for students, and the group paid $$39.50$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$13.00$ each for teachers and $$9.50$ each for students, and the group paid $$134.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+2y = 39.5}$ ${13x+9.5y = 134}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-13x-4y = -79}$ ${13x+9.5y = 134}$ Add the top and bottom equations together. $ 5.5y = 55 $ $ y = \dfrac{55}{5.5}$ ${y = 10}$ Now that you know ${y = 10}$ , plug it back into $ {6.5x+2y = 39.5}$ to find $x$ ${6.5x + 2}{(10)}{= 39.5}$ $6.5x+20 = 39.5$ $6.5x = 19.5$ $x = \dfrac{19.5}{6.5}$ ${x = 3}$ You can also plug ${y = 10}$ into $ {13x+9.5y = 134}$ and get the same answer for $x$ ${13x + 9.5}{(10)}{= 134}$ ${x = 3}$ There were $3$ teachers and $10$ students on the field trips.